A166634 Totally multiplicative sequence with a(p) = 4*(p-1) for prime p.

1, 4, 8, 16, 16, 32, 24, 64, 64, 64, 40, 128, 48, 96, 128, 256, 64, 256, 72, 256, 192, 160, 88, 512, 256, 192, 512, 384, 112, 512, 120, 1024, 320, 256, 384, 1024, 144, 288, 384, 1024, 160, 768, 168, 640, 1024, 352, 184, 2048, 576, 1024

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = (4*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)-1)^e(k).

a(n) = A165825(n) * A003958(n) = 4^bigomega(n) * A003958(n) = 4^A001222(n) * A003958(n).

Mathematica

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] :=
DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*4^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (4*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

Prog

(PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = 4*(f[k, 1]-1)); factorback(f); } \\ Michel Marcus, May 20 2016

Crossrefs

Cf. A001222, A003958, A165825, A166633.

Sequence in context: A312762 A312763 A312764 * A189007 A242349 A309521

Adjacent sequences: A166631 A166632 A166633 * A166635 A166636 A166637

Keyword

nonn,easy,mult

Author

Jaroslav Krizek, Oct 18 2009

Status

approved